The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type

The error bounds of Gauss-Lobatto quadratures for weights of Bernstein-Szegö type

“…For one class of Bernstein-Szegő weights for analytic integrands, explicit error bounds were obtained in [12], [13], [14] for Gaussian quadrature formulas, and in [9] and [4] for Gauss-Radau and Gauss-Kronrod quadrature formulas. In this paper, we continue with the analogous analysis for Gauss-Lobatto quadrature and obtain the error bounds.…”

“…Denote the (monic) orthogonal polynomials relative to the weight functions (2) and 3 . Then the remainder term in (1) admits the contour integral representation (4) R…”

“…Recently, we found some sufficient conditions for the modulus of the kernel to be maximized on one of the axes for each ρ large enough (see [12], [13], [14]) for Gaussian quadrature formulas relative to the weight functions of Bernstein-Szegő type. This issue has been examined for the Gauss-Radau quadratures in [9], and for the Gauss-Kronrod quadratures in [4]. A more general class of the Bernstein-Szegő weight functions has been considered in [8].…”

Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

“…Section 3 describes L ∞ -error bounds for Gauss-Radau rules, and in Section 4 for Gauss-Kronrod quadrature formulas for the Bernstein-Szegő weight functions; cf. [80] and [9]. Additionally, we shall prove some statements which we use in those papers.…”

“…Our first published paper on this topic was Milovanović and Spalević [39]. In the meanwhile, we and our collaborators published a number of related papers; see [9], [32]- [34], [39]- [46], [37], [49]- [55], [69], [73]- [81], [86]- [92]. We considered in our papers mainly error bounds and estimates of the type L ∞ , L 1 , and the ones based on expanding the remainder term into a series for the quadrature formulas with multiple and simple nodes of Gaussian type, including Kronrod extensions, Radau, and Lobatto modifications, mainly with the specific weight functions such as the generalized and ordinary Chebyshev weights (see [5,70]), the Gori-Micchelli weights (see [24]), the Bernstein-Szegő weight functions (see [19]), and some of their modifications; see [17].…”